Hermite-Hadamard Inequalities Research Articles

Some Novel Estimates of Hermite–Hadamard and Jensen Type Inequalities for (h1,h2)-Convex Functions Pertaining to Total Order Relation

There are different types of order relations that are associated with interval analysis for determining integral inequalities. The purpose of this paper is to connect the inequalities terms to total order relations, often called (CR)-order. In contrast to classical interval-order relations, total order relations are quite different and novel in the literature and are calculated as ω=⟨ωc,ωr⟩=⟨ω¯+ω̲2,ω¯−ω̲2⟩. A major benefit of total order relations is that they produce more efficient results than other order relations. This study introduces the notion of CR-(h1,h2)-convex function using total order relations. Center and Radius order relations are a powerful tool for studying inequalities based on their properties and widespread application. Using this novel notion, we first developed some variants of Hermite–Hadamard inequality and then constructed Jensen inequality. Based on the results, this new concept is extremely useful in connection with a variety of inequalities. There are many new and well-known convex functions unified by this type of convexity. These results will stimulate further research on inequalities for fractional interval-valued functions and fuzzy interval-valued functions, as well as the optimization problems associated with them. For the purpose of verifying our main findings, we provide some nontrivial examples.

New quantum inequalities of Hermite-Hadamard type via Green function

In this study, the Hermite-Hadamard inequality for q^{kappa_2}-integrals is demonstrated by a new method called the Green Function Technique. For this purpose, we first obtain certain identities. Then, by using these identities, we establish many new inequalities for functions whose second derivative is convex, monotone and concave in absolute value.

Refined Hermite–Hadamard Inequalities and Some Norm Inequalities

It is well known that the Hermite–Hadamard inequality (called the HH inequality) refines the definition of convexity of function f(x) defined on [a,b] by using the integral of f(x) from a to b. There are many generalizations or refinements of HH inequality. Furthermore HH inequality has many applications to several fields of mathematics, including numerical analysis, functional analysis, and operator inequality. Recently, we gave several types of refined HH inequalities and obtained inequalities which were satisfied by weighted logarithmic means. In this article, we give an N-variable Hermite–Hadamard inequality and apply to some norm inequalities under certain conditions. As applications, we obtain several inequalities which are satisfied by means defined by symmetry. Finally, we obtain detailed integral values.

A new version of ( p,q ) -Hermite–Hadamard’s midpoint and trapezoidal inequalities via special operators in ( p,q ) -calculus

In this paper, we conduct a research on a new version of the ( p,q ) -Hermite–Hadamard inequality for convex functions in the framework of postquantum calculus. Moreover, we derive several estimates for (p,q)-midpoint and (p,q)-trapezoidal inequalities for special ( p,q ) -differentiable functions by using the notions of left and right (p,q ) -derivatives. Our newly obtained inequalities are extensions of some existing inequalities in other studies. Lastly, we consider some mathematical examples for some (p,q)-functions to confirm the correctness of newly established results.

Some Hermite–Hadamard type inequalities for GA‐s‐convex functions in the fourth sense

In the present manuscript, we define the concept of ‐ ‐convex functions in the fourth sense. We obtain the Hermite–Hadamard inequality for the newly introduced class of functions. We also establish some inequalities of the Hermite–Hadamard type for functions whose first derivative in absolute value at certain powers is a ‐ ‐convex function in the fourth sense. In the last, some applications to special means of real numbers are given.

Further refinements and reverses of the Hermite–Hadamard inequalities for multivariate operator convex maps

Further refinements and reverses of the Hermite–Hadamard inequalities for multivariate operator convex maps

Some Fejér-Type Inequalities for Generalized Interval-Valued Convex Functions

The goal of this study is to create new variations of the well-known Hermite–Hadamard inequality (HH-inequality) for preinvex interval-valued functions (preinvex I-V-Fs). We develop several additional inequalities for the class of functions whose product is preinvex I-V-Fs. The findings described here would be generalizations of those found in previous studies. Finally, we obtain the Hermite–Hadamard–Fejér inequality with the support of preinvex interval-valued functions. Some new and classical special cases are also obtained. Moreover, some nontrivial examples are given to check the validity of our main results.

Some General Fractional Integral Inequalities Involving LR–Bi-Convex Fuzzy Interval-Valued Functions

The main objective of this paper is to introduce a new class of convexity called left-right–bi-convex fuzzy interval-valued functions. We study this class from the perspective of fractional Hermite–Hadamard inequalities, involving a new fractional integral called the left-right–AB fractional integral. We discuss several special cases that demonstrate that our results are quite unifying. We provide non-trivial numerical examples regarding special means for positive real numbers in order to check the validity of our outcomes.

Generalized Hermite-Hadamard Inequalities on Discrete Time Scales

This paper is concerned with some new Hermite-Hadamard inequalities on two types of time scales, Z and Nc,h. Based on the substitution rules, we first prove the discrete Hermite-Hadamard inequalities on Z relating to the midpoint a+b2 and extend them to discrete fractional forms. In addition, by using traditional methods, we prove discrete Hermite-Hadamard inequalities on Nc,h and explore the corresponding fractional inequalities involving the nabla h-fractional sums. Finally, two examples are given to illustrate the obtained results.

Some New Refinements of Trapezium-Type Integral Inequalities in Connection with Generalized Fractional Integrals

The main objective of this article is to introduce a new notion of convexity, i.e., modified exponential type convex function, and establish related fractional inequalities. To strengthen the argument of the paper, we introduce two new lemmas as auxiliary results and discuss some algebraic properties of the proposed notion. Considering a generalized fractional integral operator and differentiable mappings, whose initial absolute derivative at a given power is a modified exponential type convex, various improvements of the Hermite–Hadamard inequality are presented. Thanks to the main results, some generalizations about the earlier findings in the literature are recovered.

New Hermite–Hadamard Type Inequalities in Connection with Interval-Valued Generalized Harmonically (h1,h2)-Godunova–Levin Functions

As is known, integral inequalities related to convexity have a close relationship with symmetry. In this paper, we introduce a new notion of interval-valued harmonically m,h1,h2-Godunova–Levin functions, and we establish some new Hermite–Hadamard inequalities. Moreover, we show how this new notion of interval-valued convexity has a close relationship with many existing definitions in the literature. As a result, our theory generalizes many published results. Several interesting examples are provided to illustrate our results.

On New Fractional Version of Generalized Hermite-Hadamard Inequalities

In this study, we establish a novel version of Hermite-Hadamard inequalities through neoteric generalized Riemann-Liouville fractional integrals (RLFIs). For functions with the convex absolute values of derivatives, we create a variety of midpoint and trapezoid form inequalities, including the generalized RLFIs. Moreover, multiple fractional inequalities can be produced as special cases of the findings of this study.

Hermite–Hadamard, Fejér and Pachpatte-Type Integral Inequalities for Center-Radius Order Interval-Valued Preinvex Functions

The objective of this manuscript is to establish a link between the concept of inequalities and Center-Radius order functions, which are intriguing due to their properties and widespread use. We introduce the notion of the CR (Center-Radius)-order interval-valued preinvex function with the help of a total order relation between two intervals. Furthermore, we discuss some properties of this new class of preinvexity and show that the new concept unifies several known concepts in the literature and also gives rise to some new definitions. By applying these new definitions, we have amassed many classical and novel special cases that serve as applications of the key findings of the manuscript. The computations of cr-order intervals depend upon the following concept B=⟨Bc,Br⟩=⟨B¯+B̲2,B¯−B̲2⟩. Then, for the first time, inequalities such as Hermite–Hadamard, Pachpatte, and Fejér type are established for CR-order in association with the concept of interval-valued preinvexity. Some numerical examples are given to validate the main results. The results confirm that this new concept is very useful in connection with various inequalities. A fractional version of the Hermite–Hadamard inequality is also established to show how the presented results can be connected to fractional calculus in future developments. Our presented results will motivate further research on inequalities for fractional interval-valued functions, fuzzy interval-valued functions, and their associated optimization problems.

New Hermite–Hadamard Inequalities for Convex Fuzzy-Number-Valued Mappings via Fuzzy Riemann Integrals

This study uses fuzzy order relations to examine Hermite–Hadamard inequalities (

Some integral inequalities via fractional derivatives

In this work, we obtain new versions of the Hermite-Hadamard Inequality via generalized fractional derivatives, which differentiates our work from previous known ones, which use integral operators. Several known results from the literature are particular cases of ours, demonstrating the breadth and generality of our conclusions.

Hermite–Hadamard-Type Inequalities Involving Harmonically Convex Function via the Atangana–Baleanu Fractional Integral Operator

Fractional integrals and inequalities have recently become quite popular and have been the prime consideration for many studies. The results of many different types of inequalities have been studied by launching innovative analytical techniques and applications. These Hermite–Hadamard inequalities are discovered in this study using Atangana–Baleanu integral operators, which provide both practical and powerful results. In this paper, a symmetric study of integral inequalities of Hermite–Hadamard type is provided based on an identity proved for Atangana–Baleanu integral operators and using functions whose absolute value of the second derivative is harmonic convex. The proven Hermite–Hadamard-type inequalities have been observed to be valid for a choice of any harmonic convex function with the help of examples. Moreover, fractional inequalities and their solutions are applied in many symmetrical domains.

Some Hermite-Hadamard Type Inequalities via Katugampola Fractional for pq-Convex on the Interval-Valued Coordinates

In this paper, we established the Hermite-Hadamard inequalities via Katugampola fractional. Meanwhile, interval analysis is a particular case of set-interval analysis. We established the fractional inequalities and these results are an extension of a previous research.

The New Versions of Hermite–Hadamard Inequalities for Pre-invex Fuzzy-Interval-Valued Mappings via Fuzzy Riemann Integrals

In this study, we use the fuzzy order relation to show some novel variants of Hermite–Hadamard inequalities for pre-invex fuzzy-interval-valued mappings (F-I∙V-Ms), which we term fuzzy-interval Hermite–Hadamard inequalities and fuzzy-interval Hermite–Hadamard–Fejér inequalities. This fuzzy order relation is defined as the level of the fuzzy-interval space by the Kulisch–Miranker order relation. There are also some new exceptional instances mentioned. The theory proposed in this research is shown with practical examples that demonstrate its usefulness. This paper's approaches and methodologies might serve as a springboard for future study in this field.

MϕA-h-Convexity and Hermite-Hadamard Type Inequalities

We investigate a family of MϕA-h-convex functions, give some properties of it and several inequalities which are counterparts to the classical inequalities such as the Jensen inequality and the Schur inequality. We give the weighted Hermite-Hadamard inequalities for an MϕA-h-convex function and several estimations for the product of two functions.

Sharp bounds for the Tsallis relative operator entropy

We give the sharp and refined bounds for the Tsallis relative operator entropy by using the improved Hermite–Hadamard inequality. As an application of our result, we give new bounds for the generalized relative operator entropy. We refine the bounds of the relative operator entropy announced by Fujii and Kamei and provide the precise and sharp bounds.